--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/EquivClass.ML Tue Aug 16 19:06:14 1994 +0200
@@ -0,0 +1,268 @@
+(* Title: ZF/EquivClass.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+For EquivClass.thy. Equivalence relations in Zermelo-Fraenkel Set Theory
+*)
+
+val RSLIST = curry (op MRS);
+
+open EquivClass;
+
+(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
+
+(** first half: equiv(A,r) ==> converse(r) O r = r **)
+
+goalw EquivClass.thy [trans_def,sym_def]
+ "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
+by (fast_tac (ZF_cs addSEs [converseD,compE]) 1);
+val sym_trans_comp_subset = result();
+
+goalw EquivClass.thy [refl_def]
+ "!!A r. [| refl(A,r); r <= A*A |] ==> r <= converse(r) O r";
+by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 1);
+val refl_comp_subset = result();
+
+goalw EquivClass.thy [equiv_def]
+ "!!A r. equiv(A,r) ==> converse(r) O r = r";
+by (rtac equalityI 1);
+by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
+ ORELSE etac conjE 1));
+val equiv_comp_eq = result();
+
+(*second half*)
+goalw EquivClass.thy [equiv_def,refl_def,sym_def,trans_def]
+ "!!A r. [| converse(r) O r = r; domain(r) = A |] ==> equiv(A,r)";
+by (etac equalityE 1);
+by (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r" 1);
+by (safe_tac ZF_cs);
+by (fast_tac (ZF_cs addSIs [converseI] addIs [compI]) 3);
+by (ALLGOALS (fast_tac
+ (ZF_cs addSIs [converseI] addIs [compI] addSEs [compE])));
+by flexflex_tac;
+val comp_equivI = result();
+
+(** Equivalence classes **)
+
+(*Lemma for the next result*)
+goalw EquivClass.thy [trans_def,sym_def]
+ "!!A r. [| sym(r); trans(r); <a,b>: r |] ==> r``{a} <= r``{b}";
+by (fast_tac ZF_cs 1);
+val equiv_class_subset = result();
+
+goalw EquivClass.thy [equiv_def]
+ "!!A r. [| equiv(A,r); <a,b>: r |] ==> r``{a} = r``{b}";
+by (safe_tac (subset_cs addSIs [equalityI, equiv_class_subset]));
+by (rewrite_goals_tac [sym_def]);
+by (fast_tac ZF_cs 1);
+val equiv_class_eq = result();
+
+val prems = goalw EquivClass.thy [equiv_def,refl_def]
+ "[| equiv(A,r); a: A |] ==> a: r``{a}";
+by (cut_facts_tac prems 1);
+by (fast_tac ZF_cs 1);
+val equiv_class_self = result();
+
+(*Lemma for the next result*)
+goalw EquivClass.thy [equiv_def,refl_def]
+ "!!A r. [| equiv(A,r); r``{b} <= r``{a}; b: A |] ==> <a,b>: r";
+by (fast_tac ZF_cs 1);
+val subset_equiv_class = result();
+
+val prems = goal EquivClass.thy
+ "[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> <a,b>: r";
+by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
+val eq_equiv_class = result();
+
+(*thus r``{a} = r``{b} as well*)
+goalw EquivClass.thy [equiv_def,trans_def,sym_def]
+ "!!A r. [| equiv(A,r); x: (r``{a} Int r``{b}) |] ==> <a,b>: r";
+by (fast_tac ZF_cs 1);
+val equiv_class_nondisjoint = result();
+
+goalw EquivClass.thy [equiv_def] "!!A r. equiv(A,r) ==> r <= A*A";
+by (safe_tac ZF_cs);
+val equiv_type = result();
+
+goal EquivClass.thy
+ "!!A r. equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A";
+by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
+ addDs [equiv_type]) 1);
+val equiv_class_eq_iff = result();
+
+goal EquivClass.thy
+ "!!A r. [| equiv(A,r); x: A; y: A |] ==> r``{x} = r``{y} <-> <x,y>: r";
+by (fast_tac (ZF_cs addIs [eq_equiv_class, equiv_class_eq]
+ addDs [equiv_type]) 1);
+val eq_equiv_class_iff = result();
+
+(*** Quotients ***)
+
+(** Introduction/elimination rules -- needed? **)
+
+val prems = goalw EquivClass.thy [quotient_def] "x:A ==> r``{x}: A/r";
+by (rtac RepFunI 1);
+by (resolve_tac prems 1);
+val quotientI = result();
+
+val major::prems = goalw EquivClass.thy [quotient_def]
+ "[| X: A/r; !!x. [| X = r``{x}; x:A |] ==> P |] \
+\ ==> P";
+by (rtac (major RS RepFunE) 1);
+by (eresolve_tac prems 1);
+by (assume_tac 1);
+val quotientE = result();
+
+goalw EquivClass.thy [equiv_def,refl_def,quotient_def]
+ "!!A r. equiv(A,r) ==> Union(A/r) = A";
+by (fast_tac eq_cs 1);
+val Union_quotient = result();
+
+goalw EquivClass.thy [quotient_def]
+ "!!A r. [| equiv(A,r); X: A/r; Y: A/r |] ==> X=Y | (X Int Y <= 0)";
+by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
+by (assume_tac 1);
+by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
+by (fast_tac ZF_cs 1);
+val quotient_disj = result();
+
+(**** Defining unary operations upon equivalence classes ****)
+
+(** These proofs really require as local premises
+ equiv(A,r); congruent(r,b)
+**)
+
+(*Conversion rule*)
+val prems as [equivA,bcong,_] = goal EquivClass.thy
+ "[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r``{a}. b(x)) = b(a)";
+by (cut_facts_tac prems 1);
+by (rtac ([refl RS UN_cong, UN_constant] MRS trans) 1);
+by (etac equiv_class_self 2);
+by (assume_tac 2);
+by (rewrite_goals_tac [equiv_def,sym_def,congruent_def]);
+by (fast_tac ZF_cs 1);
+val UN_equiv_class = result();
+
+(*Resolve th against the "local" premises*)
+val localize = RSLIST [equivA,bcong];
+
+(*type checking of UN x:r``{a}. b(x) *)
+val _::_::prems = goalw EquivClass.thy [quotient_def]
+ "[| equiv(A,r); congruent(r,b); X: A/r; \
+\ !!x. x : A ==> b(x) : B |] \
+\ ==> (UN x:X. b(x)) : B";
+by (cut_facts_tac prems 1);
+by (safe_tac ZF_cs);
+by (rtac (localize UN_equiv_class RS ssubst) 1);
+by (REPEAT (ares_tac prems 1));
+val UN_equiv_class_type = result();
+
+(*Sufficient conditions for injectiveness. Could weaken premises!
+ major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
+*)
+val _::_::prems = goalw EquivClass.thy [quotient_def]
+ "[| equiv(A,r); congruent(r,b); \
+\ (UN x:X. b(x))=(UN y:Y. b(y)); X: A/r; Y: A/r; \
+\ !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |] \
+\ ==> X=Y";
+by (cut_facts_tac prems 1);
+by (safe_tac ZF_cs);
+by (rtac (equivA RS equiv_class_eq) 1);
+by (REPEAT (ares_tac prems 1));
+by (etac box_equals 1);
+by (REPEAT (ares_tac [localize UN_equiv_class] 1));
+val UN_equiv_class_inject = result();
+
+
+(**** Defining binary operations upon equivalence classes ****)
+
+
+goalw EquivClass.thy [congruent_def,congruent2_def,equiv_def,refl_def]
+ "!!A r. [| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))";
+by (fast_tac ZF_cs 1);
+val congruent2_implies_congruent = result();
+
+val equivA::prems = goalw EquivClass.thy [congruent_def]
+ "[| equiv(A,r); congruent2(r,b); a: A |] ==> \
+\ congruent(r, %x1. UN x2:r``{a}. b(x1,x2))";
+by (cut_facts_tac (equivA::prems) 1);
+by (safe_tac ZF_cs);
+by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
+by (assume_tac 1);
+by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
+ congruent2_implies_congruent]) 1);
+by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
+by (fast_tac ZF_cs 1);
+val congruent2_implies_congruent_UN = result();
+
+val prems as equivA::_ = goal EquivClass.thy
+ "[| equiv(A,r); congruent2(r,b); a1: A; a2: A |] \
+\ ==> (UN x1:r``{a1}. UN x2:r``{a2}. b(x1,x2)) = b(a1,a2)";
+by (cut_facts_tac prems 1);
+by (asm_simp_tac (ZF_ss addsimps [equivA RS UN_equiv_class,
+ congruent2_implies_congruent,
+ congruent2_implies_congruent_UN]) 1);
+val UN_equiv_class2 = result();
+
+(*type checking*)
+val prems = goalw EquivClass.thy [quotient_def]
+ "[| equiv(A,r); congruent2(r,b); \
+\ X1: A/r; X2: A/r; \
+\ !!x1 x2. [| x1: A; x2: A |] ==> b(x1,x2) : B |] \
+\ ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B";
+by (cut_facts_tac prems 1);
+by (safe_tac ZF_cs);
+by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
+ congruent2_implies_congruent_UN,
+ congruent2_implies_congruent, quotientI]) 1));
+val UN_equiv_class_type2 = result();
+
+
+(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
+ than the direct proof*)
+val prems = goalw EquivClass.thy [congruent2_def,equiv_def,refl_def]
+ "[| equiv(A,r); \
+\ !! y z w. [| w: A; <y,z> : r |] ==> b(y,w) = b(z,w); \
+\ !! y z w. [| w: A; <y,z> : r |] ==> b(w,y) = b(w,z) \
+\ |] ==> congruent2(r,b)";
+by (cut_facts_tac prems 1);
+by (safe_tac ZF_cs);
+by (rtac trans 1);
+by (REPEAT (ares_tac prems 1
+ ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
+val congruent2I = result();
+
+val [equivA,commute,congt] = goal EquivClass.thy
+ "[| equiv(A,r); \
+\ !! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y); \
+\ !! y z w. [| w: A; <y,z>: r |] ==> b(w,y) = b(w,z) \
+\ |] ==> congruent2(r,b)";
+by (resolve_tac [equivA RS congruent2I] 1);
+by (rtac (commute RS trans) 1);
+by (rtac (commute RS trans RS sym) 3);
+by (rtac sym 5);
+by (REPEAT (ares_tac [congt] 1
+ ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
+val congruent2_commuteI = result();
+
+(***OBSOLETE VERSION
+(*Rules congruentI and congruentD would simplify use of rewriting below*)
+val [equivA,ZinA,congt,commute] = goalw EquivClass.thy [quotient_def]
+ "[| equiv(A,r); Z: A/r; \
+\ !!w. [| w: A |] ==> congruent(r, %z.b(w,z)); \
+\ !!x y. [| x: A; y: A |] ==> b(y,x) = b(x,y) \
+\ |] ==> congruent(r, %w. UN z: Z. b(w,z))";
+val congt' = rewrite_rule [congruent_def] congt;
+by (cut_facts_tac [ZinA,congt] 1);
+by (rewtac congruent_def);
+by (safe_tac ZF_cs);
+by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
+by (assume_tac 1);
+by (asm_simp_tac (ZF_ss addsimps [congt RS (equivA RS UN_equiv_class)]) 1);
+by (rtac (commute RS trans) 1);
+by (rtac (commute RS trans RS sym) 3);
+by (rtac sym 5);
+by (REPEAT (ares_tac [congt' RS spec RS spec RS mp] 1));
+val congruent_commuteI = result();
+***)